effective hironaka resolution and its complexity...

EFFECTIVE HIRONAKA RESOLUTION AND ITS COMPLEXITY

EDWARD BIERSTONE, DIMA GRIGORIEV, PIERRE MILMAN, JAROS LAW W LODARCZYK

Abstract. Building upon works of Hironaka, Bierstone-Milman, Villamayor and W lodarczyk we give apri-ori estimate for the complexity of the simplified Hironaka algorithm.

Dedicated to Professor Heisuke Hironaka on the occasion of his 80th birthday

Contents

0. Introduction 1

1. Formulation of the Hironaka resolution theorems 2

2. Marked ideals, coefficient ideals and hypersurfaces of maximal contact 3

3. Resolution algorithm 10

4. Complexity bounds on a blow-up 13

5. Bounds on multiplicities and degrees of coefficient ideals 18

6. Complexity bound of the resolution algorithm in terms of Grzegorczyk’s classes 21

References 26

0. Introduction

In the present paper we discuss the complexity of the Hironaka theorem on resolution of singularities of amarked ideal. Recall that approach to the problem of embedded resolution was originated by Hironaka (see[31]) and later developed and simplified by Bierstone-Milman (see [7], [8], [9]) and Villamayor (see [46], [47]),and others. In particular, we also use some elements from the recent development by Kollar ([37]).

It seems easier to estimate the complexity of the resolution algorithm from the recursive descriptions inW lodarczyk [51] or Bierstone-Milman [12] than from the earlier iterative versions. The algorithms in [51]and [12] (or [10]) lead to identical blowing-up sequences; whether one proof is preferable to the other ispartly a matter of taste. In this article, we estimate the complexity of the “weak-strong desingularization”algorithm (see Section 1) using the construction of [51], though [12] could also be used (see Remark in thissection below). In a subsequent paper, we plan to use [12] to give a comparable complexity estimate forthe algorithm of “strong desingularization” (where the centres of blowing up are smooth subvarieties of thesuccessive strict transforms).

The basic question which arises is in what terms to estimate apriori the complexity? We recall (see e. g.[50], [26]) that the complexity is usually measured as a function on the bit-size of the input. In particular,in the paper we study varieties and ideals which are represented by families of polynomials with integercoefficients, and the vector of all these coefficients (for an initial variety and an ideal) is treated as an input.Hironaka’s algorithm consists of many steps of elementary calculations, but they are organized in several(nested) recursions when the resolution of an object (a variety or a marked ideal, see below) is reduced toresolutions of suitable objects with less values of appropriate parameters (like dimension or multiplicity). Itis instructive to represent Hironaka algorithm as a tree to each of its nodes a corresponds a marked ideal. Themarked ideals which correspond to child nodes of a have either less multiplicity of an ideal or less dimensionof a variety. An initial marked ideal corresponds to the root of the tree. The depth of the tree is bounded

Date: May 3, 2010.Bierstone’s and Milman’s research was partially supported by NSERC Discovery Grants OGP 0009070 and OGP 008949.W lodarczyk was supported in part by NSF grant DMS-0100598 and Polish KBN grant 2 P03 A 005 16.

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2 EDWARD BIERSTONE, DIMA GRIGORIEV, PIERRE MILMAN, JAROS LAW W LODARCZYK

by 2 ·m where m denotes the dimension of the initial variety, while the number of the nested recursions doesnot exceed m+3. It appears that just the number of nested recursions brings the overwhelming contributioninto the complexity of the Hironaka’s algorithm.

That is why as a relevant language for expressing a complexity bound we have chosen the Grzegorczyk’sclasses E l, l ≥ 0 [27], [50] which consists of (integer) functions whose construction requires l nested primitiverecursions. Classes E l, l ≥ 0 provide a hierarchy of the set of all primitive-recursive functions ∪l<∞E l. Inparticular, E2 contains all the (integer) polynomials and E3 contains all finite compositions of the exponentialfunction. Thus, the principal complexity result of the paper (Theorem 6.4.2) states that the complexity ofresolution of an ideal on m-dimensional variety is bounded by a function from class Em+3. We mention alsothat the complexity of (much simpler from the pure mathematical point of view) the Hilbert’s Idealbasissatzfor polynomial ideals in n variables belongs to class En+1 (cf. [42], [44], where the latter was formulatedin different languages), and moreover number n + 1 is sharp. This shows that these two quite differentalgorithmic problems have a common feature in the recursion on the dimension which mainly determinestheir complexities.

Remark. The main differences between the proofs in [12] and [51] come from the notions of derivativeideal that are used ([12] uses only derivatives that preserve the ideal of the exceptional divisor) and frompassage to a “homogenized ideal” in [51] (see §2.8). The latter has the advantage that any two maximalcontact hypersurfaces for the homogenized ideal are related by an automorphism, while [12] provides astronger version of functoriality that is needed for strong desingularization. Since [12] does not involvehomogenization, certain complexity estimates can be improved (see Remark after Corollary 5.0.10), althoughthe overall Grzegorczyk complexity class Em+3 is unchanged.

We mention that in [45] a polynomial complexity algorithm for resolution of a curve is exhibited.

Below in Section 1 we formulate the results on the canonical principalization of a sheaf of ideals and onthe embedded desingularization. In Section 2 we give basic definitions like marked ideals, hypersurfaces ofmaximal contact, coefficient ideals and formulate their properties (the omitted proofs one can find in [51]).In Section 3 we describe the resolution algorithm. In Section 4 we provide bounds on the degrees and on thenumber of polynomials which describe a single blow-up. In Section 5 we give some auxiliary bounds: on themultiplicity of an ideal in terms of degrees of describing polynomials, on the degree of a hypersurface of themaximal contact, and on the number of generators and on their degrees of the coefficient ideal. Finally inSection 6 we estimate the complexity of the resolution algorithm in terms of the Grzegorczyk’s classes (theirdefinition is also provided in Section 6).

0.1. Acknowledgements. We heartily thank Max Planck Institut fuer Mathematik, Bonn for their warmhospitality.

1. Formulation of the Hironaka resolution theorems

All algebraic varieties in this paper are defined over a ground field of characteristic zero. The assumptionof characteristic zero is only needed for the local existence of a hypersurface of maximal contact (Lemma2.6.4).

We give a proof of the following Hironaka Theorems (see [31]):

(1) Canonical Principalization

Theorem 1.0.1. Let I be a sheaf of ideals on a smooth algebraic variety X. There exists a princi-palization of I, that is, a sequence

X = X0σ1←− X1

σ2←− X2 ←− . . .←− Xi ←− . . .←− Xr = X

of blow-ups σi : Xi−1 ← Xi of smooth centers Ci−1 ⊂ Xi−1 such that(a) The exceptional divisor Ei of the induced morphism σi = σ1 . . . σi : Xi → X has only simple

normal crossings and Ci has simple normal crossings with Ei.

(b) The total transform σr∗(I) is the ideal of a simple normal crossing divisor E which is a naturalcombination of the irreducible components of the divisor Er.

The morphism (X, I) → (X, I) defined by the above principalization commutes with smooth mor-phisms and embeddings of ambient varieties. It is equivariant with respect to any group action notnecessarily preserving the ground field K.

EFFECTIVE HIRONAKA RESOLUTION AND ITS COMPLEXITY 3

(2) Weak-Strong Hironaka Embedded Desingularization

Theorem 1.0.2. Let Y be a subvariety of a smooth variety X over a field of characteristic zero.There exists a sequence

X0 = Xσ1←− X1

σ2←− X2 ←− . . .←− Xi ←− . . .←− Xr = X

of blow-ups σi : Xi−1 ←− Xi of smooth centers Ci−1 ⊂ Xi−1 such that(a) The exceptional divisor Ei of the induced morphism σi = σ1 . . . σi : Xi → X has only simple

normal crossings and Ci has simple normal crossings with Ei.(b) Let Yi ⊂ Xi be the strict transform of Y . All centers Ci are disjoint from the set Reg(Y ) ⊂ Yi

of points where Y (not Yi) is smooth (and are not necessarily contained in Yi).

(c) The strict transform Y := Yr of Y is smooth and has only simple normal crossings with theexceptional divisor Er.

(d) The morphism (X,Y ) ← (X, Y ) defined by the embedded desingularization commutes withsmooth morphisms and embeddings of ambient varieties. It is equivariant with respect to anygroup action not necessarily preserving the ground K.

(3) Canonical Resolution of Singularities

Theorem 1.0.3. Let Y be an algebraic variety over a field of characteristic zero.

There exists a canonical desingularization of Y that is a smooth variety Y together with a proper

birational morphism resY : Y → Y which is functorial with respect to smooth morphisms. For any

smooth morphism φ : Y ′ → Y there is a natural lifting φ : Y ′ → Y which is a smooth morphism.

In particular resY : Y → Y is an isomorphism over the nonsingular part of Y . Moreover resY isequivariant with respect to any group action not necessarily preserving the ground field.

Remark. Note that the blow-up of codimension one components is an isomorphism. However it defines anontrivial transformation of marked ideals. In the actual desingularization process this kind of blow-up mayoccur for some marked ideals induced on subvarieties of ambient varieties. Though they define isomorphismsof those subvarieties they determine blow-ups of ambient varieties which are not isomorphisms.

Remarks. (1) By the exceptional divisor of the blow-up σ : X ′ → X with a smooth center C we mean theinverse image E := σ−1(C) of the center C. By the exceptional divisor of the composite of blow-upsσi with smooth centers Ci−1 we mean the union of the strict transforms of the exceptional divisorsof σi. This definition coincides with the standard definition of the exceptional set of points of thebirational morphism in the case when codim(Ci) ≥ 2 (as in Theorem 1.0.2). If codim(Ci−1) = 1 theblow-up of Ci−1 is an identical isomorphism and defines a formal operation of converting a subvarietyCi−1 ⊂ Xi−1 into a component of the exceptional divisor Ei on Xi. This formalism is convenient forthe proofs. In particular it indicates that Ci−1 identified via σi with a component of Ei has simplenormal crossings with other components of Ei.

(2) In the Theorem 1.0.2 we blow up centers of codimension ≥ 2 and both definitions coincide.

2. Marked ideals, coefficient ideals and hypersurfaces of maximal contact

We shall assume that the ground field is algebraically closed.

2.1. Resolution of marked ideals. For any sheaf of ideals I on a smooth variety X and any point x ∈ Xwe denote by

ordx(I) := maxi | I ⊂ mix

the order of I at x. (Here mx denotes the maximal ideal of x.)

Definition 2.1.1. (Hironaka (see [31], [33]), Bierstone-Milman (see [8]),Villamayor (see [46])) A marked ideal(originally a basic object of Villamayor) is a collection (X, I, E, µ), where X is a smooth variety, I is a sheafof ideals on X , µ is a nonnegative integer and E is a totally ordered collection of divisors whose irreduciblecomponents are pairwise disjoint and all have multiplicity one. Moreover the irreducible components ofdivisors in E have simultaneously simple normal crossings.

Definition 2.1.2. (Hironaka ([31], [33]), Bierstone-Milman (see [8]),Villamayor (see [46])) By the support(originally singular locus) of (X, I, E, µ) we mean

supp(X, I, E, µ) := x ∈ X | ordx(I) ≥ µ.


Remarks. (1) Sometimes for simplicity we shall represent marked ideals (X, I, E, µ) as couples (I, µ) oreven ideals I.

(2) For any sheaf of ideals I on X we have supp(I, 1) = supp(I).(3) For any marked ideals (I, µ) on X , supp(I, µ) is a closed subset of X (Lemma 2.5.2).

Definition 2.1.3. (Hironaka (see [31], [33]), Bierstone-Milman (see [8]),Villamayor (see [46])) By a resolutionof (X, I, E, µ) we mean a sequence of blow-ups σi : Xi → Xi−1 of disjoint unions of smooth centers Ci−1 ⊂Xi−1,

X0 = Xσ1←− X1

σ2←− X2σ3←− . . . Xi ←− . . .

σr←− Xr,

which defines a sequence of marked ideals (Xi, Ii, Ei, µ) where

(1) Ci ⊂ supp(Xi, Ii, Ei, µ).(2) Ci has simple normal crossings with Ei.(3) Ii = I(Di)

−µσ∗i (Ii−1), where I(Di) is the ideal of the exceptional divisor Di of σi.

(4) Ei = σci (Ei−1) ∪ Di, where σc

i (Ei−1) is the set of strict transforms of divisors in Ei−1.(5) The order on σc

i (Ei−1) is defined by the order on Ei−1 while Di is the maximal element of Ei.(6) supp(Xr, Ir, Er, µ) = ∅.

Definition 2.1.4. The sequence of morphisms which are either isomorphisms or blow-ups satisfying condi-tions (1)-(5) is called a multiple blow-up. The number of morphisms in a multiple blow-up will be called itslength.

Definition 2.1.5. An extension of a multiple blow-up (or a resolution) (Xi)0≤i≤m is a sequence (X ′j)0≤j≤m′

of blow-ups and isomorphisms X ′0 = X ′

j0 = . . . = X ′j1−1 ← X ′

j1 = . . . = X ′j2−1 ← . . . X ′

jm = . . . = X ′m′ ,

where X ′ji

= Xi.

Remarks. (1) The definition of extension arises naturally when we pass to open subsets of the consideredambient variety X .

(2) The notion of a multiple blow-up is analogous to the notions of or admissible blow-ups consideredby Hironaka, Bierstone-Milman and Villamayor.

2.2. Transforms of marked ideal and controlled transforms of functions. In the setting of the abovedefinition we shall call

(Ii, µ) := σci (Ii−1, µ)

a transform of the marked ideal or controlled transform of (I, µ). It makes sense for a single blow-up in amultiple blow-up as well as for a multiple blow-up. Let σi := σ1 . . . σi : Xi → X be a composition ofconsecutive morphisms of a multiple blow-up. Then in the above setting

(Ii, µ) = σic(I, µ).

We shall also denote the controlled transform σic(I, µ) by (I, µ)i or [I, µ]i.

The controlled transform can also be defined for local sections f ∈ I(U). Let σ : X ← X ′ be a blow-up ofa smooth center C ⊂ supp(I, µ) defining transformation of marked ideals σc(I, µ) = (I ′, µ). Let f ∈ I(U)be a section of a sheaf of ideals. Let U ′ ⊆ σ−1(U) be an open subset for which the sheaf of ideals of theexceptional divisor is generated by a function y. The function

g = y−µ(f σ) ∈ I(U ′)

is a controlled transform of f on U ′ (defined up to an invertible function). As before we extend it to anymultiple blow-up.

The following lemma shows that the notion of controlled transform is well defined.

Lemma 2.2.1. Let C ⊂ supp(I, µ) be a smooth center of the blow-up σ : X ← X ′ and let D denote theexceptional divisor. Let IC denote the sheaf of ideals defined by C. Then

(1) I ⊂ IµC .(2) σ∗(I) ⊂ (ID)µ.

Proof. (1) We can assume that the ambient variety X is affine. Let u1, . . . , uk be parameters generating ICSuppose f ∈ I \ IµC . Then we can write f =

∑α cαu

α, where either |α| ≥ µ or |α| < µ and cα 6∈ IC . By theassumption there is α with |α| < µ such that cα 6∈ IC . Take α with the smallest |α|. There is a point x ∈ C


for which cα(x) 6= 0 and in the Taylor expansion of f at x there is a term cα(x)uα. Thus ordx(I) < µ. Thiscontradicts to the assumption C ⊂ supp(I, µ).

(2) σ∗(I) ⊂ σ∗(IC)µ = (ID)µ.

2.3. Hironaka resolution principle. Our proof is based upon the following principle which can be tracedback to Hironaka and was used by Villamayor in his simplification of Hironaka’s algorithm:

(Canonical) Resolution of marked ideals (X, I, E, µ)(1)

⇓

(Canonical) Principalization of the sheaves I on X(2)

⇓

(Canonical) Weak Embedded Desingularization of subvarieties Y ⊂ X(3)

⇓

(Canonical) Desingularization(4)

(1)⇒(2) It follows immediately from the definition that a resolution of (X, I, ∅, 1) determines a princi-

palization of I. Denote by σ : X ← X the morphism defined by a resolution of (X, I, ∅, 1). The controlled

transform (I, 1) := σc(I, 1) has the empty support. Consequently, V (I) = ∅, and thus I is equal to thestructural sheaf OX . This implies that the full transform σ∗(I) is principal and generated by the sheaf ofideal of a divisor whose components are the exceptional divisors. The actual process of desingularization isoften achieved before (X, I, E, 1) has been resolved (see [51]) .

(2)⇒(3) Let Y ⊂ X be an irreducible subvariety. Assume there is a principalization of sheaves of idealsIY subject to conditions (a) and (b) from Theorem 1.0.1. Then in the course of the principalization of IYthe strict transform Yi of Y on some Xi is the center of a blow-up. At this stage Yi is nonsingular and hassimple normal crossing with the exceptional divisors.

(3)⇒(4) Every algebraic variety admits locally an embedding into an affine space. Thus we can show thatthe existence of canonical embedded desingularization independent of the embedding defines a canonicaldesingularization.

For more details see [51].

2.4. Equivalence relation for marked ideals. Let us introduce the following equivalence relation formarked ideals:

Definition 2.4.1. Let (X, I, EI , µI) and (X,J , EJ , µJ ) be two marked ideals on the smooth variety X .Then (X, I, EI , µI) ≃ (X,J , EJ , µJ ) if

(1) EI = EJ and the orders on EI and on EJ coincide.(2) supp(I, µI) = supp(J , µJ ).(3) The multiple blow-ups (Xi)i=0,...,k are the same for both marked ideals and supp(Ii, µI) = supp(Ji, µJ ).

Example 2.4.2. For any k ∈ N, (I, µ) ≃ (Ik, kµ).

Remark. The marked ideals considered in this paper satisfy a stronger equivalence condition: For any smoothmorphisms φ : X ′ → X , φ∗(I, µ) ≃ φ∗(J , µ). This condition will follow and is not added in the definition.

2.5. Ideals of derivatives. Ideals of derivatives were first introduced and studied in the resolution contextby Giraud. Villamayor developed and applied this language to his basic objects.

Definition 2.5.1. (Giraud, Villamayor) Let I be a coherent sheaf of ideals on a smooth variety X . Bythe first derivative (originally extension) D(I) of I we mean the coherent sheaf of ideals generated by allfunctions f ∈ I with their first derivatives. Then the i-th derivative Di(I) is defined to be D(Di−1(I)). If(I, µ) is a marked ideal and i ≤ µ then we define

Di(I, µ) := (Di(I), µ− i).


Recall that on a smooth variety X there is a locally free sheaf of differentials ΩX/K over K generatedlocally by du1, . . . , dun for a set of local parameters u1, . . . , un. The dual sheaf of derivations DerK(OX)is locally generated by the derivations ∂

∂ui. Immediately from the definition we observe that D(I) is a

coherent sheaf defined locally by generators fj of I and all their partial derivatives∂fj∂ui

. We see by induction

that Di(I) is a coherent sheaf defined locally by the generators fj of I and their derivatives∂|α|fj∂uα for all

multiindices α = (α1, . . . , αn), where |α| := α1 + . . . + αn ≤ i.

Lemma 2.5.2. (Giraud, Villamayor) For any i ≤ µ− 1,

supp(I, µ) = supp(Di(I), µ − i).

In particular supp(I, µ) = supp(Dµ−1(I), 1) = V (Dµ−1(I)) is a closed set.

We write (I, µ) ⊂ (J , µ) if I ⊂ J .

Lemma 2.5.3. (Giraud,Villamayor) Let (I, µ) be a marked ideal and C ⊂ supp(I, µ) be a smooth centerand r ≤ µ. Let σ : X ← X ′ be a blow-up at C. Then

σc(Dr(I, µ)) ⊆ Dr(σc(I, µ)).

Proof. See simple computations in [49], [51].

2.6. Hypersurfaces of maximal contact. The concept of the hypersurfaces of maximal contact is one ofthe key points of this proof. It was originated by Hironaka, Abhyankhar and Giraud and developed in thepapers of Bierstone-Milman and Villamayor.

In our terminology we are looking for a smooth hypersurface containing the supports of marked idealsand whose strict transforms under multiple blow-ups contain the supports of the induced marked ideals.Existence of such hypersurfaces allows a reduction of the resolution problem to codimension 1.

First we introduce marked ideals which locally admit hypersurfaces of maximal contact.

Definition 2.6.1. (Villamayor (see [46])) We say that a marked ideal (I, µ) is of maximal order (originallysimple basic object) if maxordx(I) | x ∈ X ≤ µ or equivalently Dµ(I) = OX .

Lemma 2.6.2. (Villamayor (see [46])) Let (I, µ) be a marked ideal of maximal order and C ⊂ supp(I, µ)be a smooth center. Let σ : X ← X ′ be a blow-up at C ⊂ supp(I, µ). Then σc(I, µ) is of maximal order.

Proof. If (I, µ) is a marked ideal of maximal order thenDµ(I) = OX . Then by Lemma 2.5.3,Dµ(σc(I, µ)) ⊃σc(Dµ(I), 0) = OX .

Lemma 2.6.3. (Villamayor (see [46])) If (I, µ) is a marked ideal of maximal order and 0 ≤ i ≤ µ thenDi(I, µ) is of maximal order.

Proof. Dµ−i(Di(I, µ)) = Dµ(I, µ) = OX .

Lemma 2.6.4. (Giraud (see [22])) Let (I, µ) be the marked ideal of maximal order. Let σ : X ← X ′ bea blow-up at a smooth center C ⊂ supp(I, µ). Let u ∈ Dµ−1(I, µ)(U) be a function such that, for anyx ∈ V (u), ordx(u) = 1. Then

(1) V (u) is smooth.(2) supp(I, µ) ∩ U ⊂ V (u)

Let U ′ ⊂ σ−1(U) ⊂ X ′ be an open set where the exceptional divisor is described by y. Let u′ := σc(u) =y−1σ∗(u) be the controlled transform of u. Then

(1) u′ ∈ Dµ−1(σc(I|U ′ , µ)).(2) V (u′) is smooth.(3) supp(I ′, µ) ∩ U ′ ⊂ V (u′)(4) V (u′) is the restriction of the strict transform of V (u) to U ′.

Proof. (1) u′ = σc(u) = u/y ∈ σc(Dµ−1(I)) ⊂ Dµ−1(σc(I)).

(2) Since u was one of the local parameters describing the center of blow-ups, u′ = u/y is a parameter,that is, a function of order one.

(3) follows from (2).


Definition 2.6.5. We shall call a function

u ∈ T (I)(U) := Dµ−1(I(U))

of multiplicity one a tangent direction of (I, µ) on U .

As a corollary from the above we obtain the following lemma:

Lemma 2.6.6. (Giraud) Let u ∈ T (I)(U) be a tangent direction of (I, µ) on U . Then for any multipleblow-up (Ui) of (I|U , µ) all the supports of the induced marked ideals supp(Ii, µ) are contained in the stricttransforms V (u)i of V (u).

Remarks. (1) Tangent directions are functions defining locally hypersurfaces of maximal contact.(2) The main problem leading to complexity of the proofs is that of noncanonical choice of the tangent

directions. We overcome this difficulty by introducing homogenized ideals.

2.7. Arithmetical operations on marked ideals. In this sections all marked ideals are defined for thesmooth variety X and the same set of exceptional divisors E. Define the following operations of additionand multiplication of marked ideals:

(1) (I, µI) + (J , µJ ) := (IµI + J µI , µIµJ ), or more generally (the operation of addition is not asso-ciative)

(I1, µ1) + . . . + (Im, µm) := (Iµ2·...·µm

1 + Iµ1µ3·...·µm

2 + . . . + Iµ1...µk−1m , µ1µ2 . . . µm).

(2) (I, µI) · (J , µJ ) := (I · J, µI + µJ ).

Lemma 2.7.1. (1) supp((I1, µ1) + . . . + (Im, µm)) = supp(I1, µ1) ∩ . . . ∩ supp(Im, µm). Moreovermultiple blow-ups (Xk) of (I1, µ1) + . . .+ (Im, µm) are exactly those which are simultaneous multipleblow-ups for all (Ij , µj) and for any k we have the equality for the controlled transforms (Ij , µI)k

(I1, µ1)k + . . . + (Im, µm)k = [(I1, µ1) + . . . + (Im, µm)]k

(2)supp(I, µI) ∩ supp(J , µJ ) ⊇ supp((I, µI) · (J , µJ )).

Moreover any simultaneous multiple blow-up Xi of both ideals (I, µI) and (J , µJ ) is a multipleblow-up for (I, µI) · (J , µJ ), and for the controlled transforms (Ik, µI) and (Jk, µJ ) we have theequality

(Ik, µI) · (Jk, µJ ) = [(I, µI) · (J , µJ )]k.

2.8. Homogenized ideals and tangent directions. Let (I, µ) be a marked ideal of maximal order. SetT (I) := Dµ−1I. By the homogenized ideal we mean

H(I, µ) := (H(I), µ) = (I +DI · T (I) + . . . +DiI · T (I)i + . . . +Dµ−1I · T (I)µ−1, µ)

Remark. A homogenized ideal features two important properties:

(1) It is equivalent to the given ideal.(2) It ”looks the same” from all possible tangent directions.

By the first property we can use the homogenized ideal to construct resolution via the Giraud Lemma 2.6.6.By the second property such a construction does not depend on the choice of tangent directions.

Lemma 2.8.1. Let (I, µ) be a marked ideal of maximal order. Then

(1) (I, µ) ≃ (H(I), µ).(2) For any multiple blow-up (Xk) of (I, µ),

(H(I), µ)k = (I, µ)k + [D(I, µ)]k · [(T (I), 1)]k + . . . [Dµ−1(I, µ)]k ·+[(T (I), 1)]µ−1k .

Although the following Lemmas 2.8.2 and 2.8.3 are used in this paper only in the case E = ∅ we formulatethem in slightly more general versions.

Lemma 2.8.2. Let (X, I, E, µ) be a marked ideal of maximal order. Assume there exist tangent direc-tions u, v ∈ T (I, µ)x = Dµ−1(I, µ)x at x ∈ supp(I, µ) which are transversal to E. Then there exists an

automorphism φuv of Xx := Spec(Ox,X) such that

(1) φ∗uv(HI)x = (HI)x.

(2) φ∗uv(E) = E.


(3) φ∗uv(u) = v.

(4) supp(I, µ) := V (T (I, µ)) is contained in the fixed point set of φ.

Proof. (0) Construction of the automorphism φuv.

Find parameters u2, . . . , un transversal to u and v such that u = u1, u2, . . . , un and v, u2, . . . , un form twosets of parameters at x and divisors in E are described by some parameters ui where i ≥ 2. Set

φuv(u1) = v, φuv(ui) = ui for i > 1.

(1) Let h := v − u ∈ T (I). For any f ∈ I,

φ∗uv(f) = f(u1 + h, u2, . . . , un) = f(u1, . . . , un) +

∂f

∂u1· h +

1

2!

∂2f

∂u21

· h2 + . . . +1

i!

∂if

∂ui1

· hi + . . .

The latter element belongs to

I + DI · T (I) + . . . +DiI · T (I)i

+ . . . +Dµ−1I · T (I)µ−1

= HI.

Hence φ∗uv(I) ⊂ HI. (2)(3) Follow from the construction.

(4) The fixed point set of φ∗uv is defined by ui = φ∗

uv(ui), i = 1, . . . , n, that is, h = 0. But h ∈ Dµ−1(I) is0 on supp(I, µ).

Lemma 2.8.3. (Glueing Lemma) Let (X, I, E, µ) be a marked ideal of maximal order for which thereexist tangent directions u, v ∈ T (I, µ) at x ∈ supp(I, µ) which are transversal to E. Then there exist etaleneighborhoods φu, φv : X → X of x = φu(x) = φv(x) ∈ X, where x ∈ X, such that

(1) φ∗u(H(I) = φ∗

v(H(I)).(2) φ∗

u(E) = φ∗v(E).

(3) φ∗u(u) = φ∗

v(v).

Set (X, I, E, µ) := φ∗u(X,H(I), E, µ) = φ∗

v(X,H(I, E, µ)).(4) For any y ∈ supp(X, I, E, µ), φu(y) = φv(y).(5) For any multiple blow-up (Xi) of (X, I, ∅, µ) the induced multiple blow-ups φ∗

u(Xi) and φ∗v(Xi) of

(X, I, E, µ) are the same (defined by the same centers).Set (X i) := φ∗

u(Xi) = φ∗v(Xi).

(6) For any yi ∈ supp(X i, Ii, Ei, µ) and the induced morphisms φui, φvi : X i → Xi, φui(yi) = φvi(yi).

Proof. (0) Construction of etale neighborhoods φu, φv : U → X.

Let U ⊂ X be an open subset for which there exist u2, . . . , un which are transversal to u and v on U suchthat u = u1, u2, . . . , un and v, u2, . . . , un form two sets of parameters on U and divisors in E are describedby some ui, where i ≥ 2. Let An be the affine space with coordinates x1, . . . , xn. Construct first etalemorphisms φ1, φ2 : U → An with

φ∗1(xi) = ui for all i and φ∗

2(x1) = v, φ∗2(xi) = ui for i > 1.

Then

X := U ×An U

is a fiber product for the morphisms φ1 and φ2. The morphisms φu, φv are defined to be the naturalprojections φu, φv : X → U such that φ1φu = φ2φv. Set

w1 := φ∗u(u) = (φ1φu)∗(x1) = (φ2φv)∗(x1) = φ∗

v(v),

wi = φ∗u(ui) = φ∗

v(ui) for i ≥ 2.

(1), (2), (3) follow from the construction

(4) Let h := v − u. By the above the morphisms φu and φv coincide on φ−1u (V (h)) = φ−1

v (V (h)).

By (4) the blow-ups of the centers C ⊂ supp(H(I)) lifts to the blow-ups at the same center φ−1u (C) =

φ−1u (C). Thus (5), (6) follow. (see [51] for details).


2.9. Coefficient ideals and Giraud Lemma. The idea of coefficient ideals was originated by Hironakaand then developed in papers of Villamayor and Bierstone-Milman. The following definition modifies andgeneralizes the definition of Villamayor.

Definition 2.9.1. Let (I, µ) be a marked ideal of maximal order. By the coefficient ideal we mean

C(I, µ) =

µ∑

i=1

(DiI, µ− i).

Remark. The coefficient ideals C(I) feature two important properties.

(1) C(I) is equivalent to I.(2) The intersection of the support of (I, µ) with any smooth subvariety S is the support of the restriction

of C(I) to S:supp(I) ∩ S = supp(C(I)|S).

Moreover this condition is persistent under relevant multiple blow-ups.

These properties allow one to control and modify the part of support of (I, µ) contained in S by applyingmultiple blow-ups of C(I)|S .

Lemma 2.9.2. C(I, µ) ≃ (I, µ).

Proof. By Lemma 2.7.1 multiple blow-ups of C(I, µ) are simultaneous multiple blow-ups of Di(I, µ) for0 ≤ i ≤ µ − 1. By Lemma 2.5.3 multiple blow-ups of (I, µ) define the multiple blow-up of all Di(I, µ).Thus multiple blow-ups of (I, µ) and C(I, µ) are the same and supp(C(I, µ))k =

⋂supp(DiI, µ − i)k =

supp(Ik, µ).

Lemma 2.9.3. Let (X, I, E, µ) be a marked ideal of maximal order. Assume that S has only simple normalcrossings with E. Then

supp(I, µ) ∩ S = supp(C(I, µ)|S).

Moreover let (Xi) be a multiple blow-up with centers Ci contained in the strict transforms Si ⊂ Xi of S.Then

(1) The restrictions σi|Si: Si → Si−1 of the morphisms σi : Xi → Xi−1 define a multiple blow-up (Si)

of C(I, µ)|S.(2) supp(Ii, µ) ∩ Si = supp[C(I, µ)|S ]i.(3) Every multiple blow-up (Si) of C(I, µ)|S defines a multiple blow-up (Xi) of (I, µ) with centers Ci

contained in the strict transforms Si ⊂ Xi of S ⊂ X.

Proof. By Lemma 2.9.2, supp(I, µ) ∩ S = supp(C(I, µ)) ∩ S ⊆ supp(C(I, µ)|S).

Let x1, . . . , xk, y1, . . . , yn−k be local parameters at x such that x1 = 0, . . . , xk = 0 describes S. Thenany function f ∈ I can be written as

f =∑

cαf (y)xα,

where cαf (y) are formal power series in yi.

Now x ∈ supp(I, µ) ∩ S iff ordx(cα) ≥ µ− |α| for all f ∈ I and |α| ≤ µ. Note that

cαf |S =

(1

α!

∂|α|(f)

∂xα

)

|S

∈ D|α|(I)|S

and consequently supp(I, µ) ∩ S =⋂

f∈I,|α|≤µ supp(cαf |S , µ− |α|) ⊃ supp(C(I, µ)|S).

The above relation is preserved by multiple blow-ups of (I, µ). For the details see [51].

Lemma 2.9.4. Let φ : X ′ → X be an etale morphism of smooth varieties and let (X, I, ∅, µ) be a markedideal. Then

(1) φ∗(D(I)) = D(φ∗(I)).(2) φ∗(H(I)) = H(φ∗(I)).(3) φ∗(C(I)) = C(φ∗(I)).

Proof. Note that for any point x ∈ X the completion φ∗x is an isomorphism. Thus φ∗

x(D(I)) = D((I)) andtherefore φ∗(D(I)) = D(φ∗(I)). (2) and (3) follow from (1).


3. Resolution algorithm

The presentation of the following Hironaka resolution algorithm builds upon Bierstone-Milman’s, Villa-mayor’s and Wlodarczyk’s algorithms which are simplifications of the original Hironaka proof. We also useKollar’s trick allowing to completely eliminate the use of invariants.

Remarks. (1) Note that the blow-up of codimension one components is an isomorphism. However itdefines a nontrivial transformation of marked ideals. The inverse image of the center is still calledthe exceptional divisor.

(2) In the actual desingularization process this kind of blow-up may occur for some marked idealsinduced on subvarieties of ambient varieties. Though they define isomorphisms of those subvarietiesthey determine blow-ups of ambient varieties which are not isomorphisms.

(3) The blow-ups of the center C which coincides with the whole variety X is an empty set. The mainfeature which characterizes is given by the restriction property:

If X is a smooth variety containing a smooth subvariety Y ⊂ X , which contains the center C ⊂ Ythen the blow-up σC,Y : Y → Y at C coincides with the strict transform of Y under the blow-up

σC,X : X → X , i.e

Y ≃ σ−1C,X(Y \ C)

Inductive assumption For any marked ideal (X, I, E, µ) such that I there is an associated resolution(Xi)0≤i≤mX

, called canonical, satisfying the following conditions:

(1) For any surjective locally etale morphism φ : X ′ → X the induced sequence (X ′i) = φ∗(Xi) is the

canonical resolution of (X ′, I ′, E′, µ) := φ∗(X, I, E, µ).(2) For any locally analytic isomophism φ : M ′ →M the induced sequence (X ′

i) = φ∗(Xi) is an extensionof the canonical resolution of (X ′, I ′, E′, µ) := φ∗(X, I, E, µ).

Proof. If I = 0 and µ > 0 then supp(X, I, µ) = X , and the blow-up of X is the empty set and thus itdefines a unique resolution. Assume that I 6= 0.

We shall use the induction on the dimension of X . If X is 0-dimensional, I 6= 0 and µ > 0 thensupp(X, I, µ) = ∅ and all resolutions are trivial.

Step 1 Resolving a marked ideal (X,J , E, µ) of maximal order.

Before performing the resolution algorithm for the marked ideal (J , µ) of maximal order in Step 1 weshall replace it with the equivalent homogenized ideal C(H(J , µ)). Resolving the ideal C(H(J , µ)) defines aresolution of (J , µ) at this step. To simplify notation we shall denote C(H(J , µ)) by (J , µ).

Step 1a Reduction to the nonboundary case. Moving supp(J , µ) and Hsα apart . For any

multiple blow-up (Xi) of (X,J , E, µ) we shall identify (for simplicity) strict transforms of E on Xi with E.

For any x ∈ Xi, let s(x) denote the number of divisors in E through x and set

si = maxs(x) | x ∈ supp(J i).

Let s = s0. By assumption the intersections of any s > s0 components of the exceptional divisorsare disjoint from supp(J , µ). Each intersection of divisors in E is locally defined by intersection of someirreducible components of these divisors. Find all intersections Hs

α, α ∈ A, of s irreducible components ofdivisors E such that supp(J , µ) ∩ Hs

α 6= ∅. By the maximality of s, the supports supp(J |Hsα

) ⊂ Hsα are

disjoint from Hsα′ , where α′ 6= α.

SetHs :=

⋃

α

Hsα, Us := X \Hs+1, Hs := Hs \Hs+1.

Then Hs ⊂ Us is a smooth closed subset Us. Moreover Hs ∩ supp(I) = Hs ∩ supp(I) is closed.

Construct the canonical resolution of J |Hs . By Lemma 2.9.3, it defines a multiple blow-up of (J , µ) suchthat

supp(J j1 , µ) ∩Hsj1 = ∅.

In particular the number of the strict tranforms of E passing through a single point of the support dropssj1 < s. Now we put s = sj1 and repeat the procedure. We continue the above process till sjk = sr = 0.

Then (Xj)0≤j≤r is a multiple blow-up of (X,J , E, µ) such that supp(J r, µ) does not intersect any divisorin E.


Therefore (Xj)0≤j≤r and further longer multiple blow-ups (Xj)0≤j≤m for any m ≥ r can be considered

as multiple blow-ups of (X,J , ∅, µ) since starting from Xr the strict transforms of E play no further rolein the resolution process since they do not intersect supp(J j , µ) for j ≥ r. We reduce the situation to the”nonboundary case”.

Step 1b. Nonboundary case

Let (Xj)0≤j≤r be the multiple blow-up of (X,J , ∅, µ) defined in Step 1a.

For any x ∈ supp(J , µ) ⊂ X find a tangent direction uα ∈ Dµ−1(J ) on some neighborhood Uα of x.Then V (uα) ⊂ Uα is a hypersurface of maximal contact. By the quasicompactness of X we can assume thatthe covering defined by Uα is finite. Let Uiα ⊂ Xi be the inverse image of Uiα and let Hiα := V (uα)i ⊂ Uiα

denote the strict transform of Hα := V (uα).

Set (see also [37])

X :=∐

Uα H :=∐

Hα ⊆ X.

The closed embeddings Hα ⊆ Uα define the closed embedding H ⊂ M of a hypersurface of maximal

contact H.

Consider the surjective etale morphism

φU : X :=∐

Uα → X

Denote by J the pull back of the ideal sheaf J via φU . The multiple blow-up (Xi)0≤i≤r of J defines a

multiple blow-up (X0≤i≤r) of J and a multiple blow-up (Hi)0≤i≤r of J|H .

Construct the canonical resolution of (Hi)r≤i≤m of the marked ideal Jr|Hron Hr. It defines, by Lemma

2.9.2, a resolution (Xr≤i≤m) of Jr and thus also a resolution (Xi)0≤i≤m of (X, J , ∅, µ). Moreover bothresolutions are related by the property

supp(Ji) = supp(Ji|Hi).

Consider a (possible) lifting of φU :

φiU : Xi :=∐

Uiα → Xi,

which is a surjective locally etale morphism. The lifting is constructed for 0 ≤ i ≤ r.

For r ≤ i ≤ m the resolution Xi is induced by the canonical resolution (Hi)r≤i≤m of J r|Hr

We show that the resolution (Xi)r≤i≤m descends to the resolution (Xi)r≤i≤m.

Let Cj0 =∐

Cj0α be the center of the blow-up σj0 : Xj0+1 → Xj0 . The closed subset Cj0α ⊂ Uj0α definesthe center of an extension of the canonical resolution (Hjα)r≤j≤m.

If Cj0α∩Uj0β 6= ∅ then by the canonicity and condition (2) of the inductive assumption, the subset Cj0αβ :=Cj0α ∩ Uj0β defines the center of an extension of of the canonical resolution Hjαβ := ((Hjα ∩ Ujβ))r≤j≤m.On the other hand Cj0βα := Cj0β ∩ Ujα defines the center of an extension of the canonical resolution((Hjβα := Hjβ ∩ Ujα))r≤j≤m.

By Glueing Lemma 2.8.3 for the tangent directions uα and uβ we find there exist etale neighborhoods

φuα, φuβ

: Uαβ → Uαβ := Uα ∩ Uβ of x = φu(x) = φv(x) ∈ X , where x ∈ X, such that

(1) φ∗uα

(J ) = φ∗uβ

(J ).

(2) φ∗uα

(E) = φ∗uβ

(E).

(3) φ−1uα

(Hjαβ) = φ−1uβ

(Hjβα).

(4) φuα(x) = φuβ

(x) for x ∈ supp(φ∗uα

(J )).

Moreover all the properties lift to the relevant etale morphisms φuαi, φuβi

: Uαβi → Uαβi. Consequently,

by canonicity φ−1uαj0

(Cj0αβ) and φ−1uβj0

(Cj0βα) define both the next center of the extension of the canonical

resolution φ−1uα

(Hj0αβ) = φ−1uβ

(Hj0βα) of φ∗uαj0(J|Hαβ

) = φ∗uβ

(J|Hβα).

Thus

φ−1uα

(Cj0αβ) = φ−1uβ

(Cj0βα),


and finally, by property (4),

Cj0αβ = Cj0βα.

Consequently Cj0 descends to the smooth closed center Cj0 =⋃Cj0α ⊂ Xj0 and the resolution (Xi)r≤i≤m

descends to the resolution (Xi)r≤i≤m.

Step 2. Resolving of marked ideals (X, I, E, µ).

For any marked ideal (X, I, E, µ) write

I =M(I)N (I),

where M(I) is the monomial part of I, that is, the product of the principal ideals defining the irreduciblecomponents of the divisors in E, and N (I) is a nonmonomial part which is not divisible by any ideal of adivisor in E. Let

ordN (I) := maxordx(N (I)) | x ∈ supp(I, µ).

Definition 3.0.5. (Hironaka, Bierstone-Milman,Villamayor, Encinas-Hauser) By the companion ideal of(I, µ) where I = N (I)M(I) we mean the marked ideal of maximal order

O(I, µ) =

(N (I), ordN (I)) + (M(I), µ− ordN (I)) if ordN (I) < µ,(N (I), ordN (I)) if ordN (I) ≥ µ.

In particular O(I, µ) = (I, µ) for ideals (I, µ) of maximal order.

Step 2a. Reduction to the monomial case by using companion ideals

By Step 1 we can resolve the marked ideal of maximal order (J , µJ ) := O(I, µ). By Lemma 2.7.1, forany multiple blow-up of O(I, µ),

supp(O(I, µ))i = supp[N (I), ordN (I)]i ∩ supp[M(I), µ− ordN (HI)]i =supp[N (I), ordN (I)]i ∩ supp(Ii, µ).

Consequently, such a resolution leads to the ideal (Ir1 , µ) such that ordN (Ir1) < ordN (I). Then we

repeat the procedure for (Ir1 , µ). We find marked ideals (Ir0 , µ) = (I, µ), (Ir1 , µ), . . . , (Irm , µ) such thatordN (I0) > ordN (Ir1) > . . . > ordN (Irm ). The procedure terminates after a finite number of steps when we

arrive at the ideal (Irm , µ) with ordN (Irm ) = 0 or with supp(Irm , µ) = ∅. In the second case we get theresolution. In the first case Irm =M(Irm) is monomial.

Step 2b. Monomial case I =M(I).

Let x1, . . . , xk define equations of the components Dx1 , . . . , D

xk ∈ E through x ∈ supp(X, I, E, µ) and I

be generated by the monomial xa1,...,ak at x. In particular

ordx(I)(x) := a1 + . . . + ak.

Let ρ(x) = Di1 , . . . , Dil ∈ Sub(E) be the maximal subset satisfying the properties

(1) ai1 + . . . + ail ≥ µ.(2) For any j = 1, . . . , l, ai1 + . . . + aij + . . . + ail < µ.

Let R(x) denote the subsets in Sub(E) satisfying the properties (1) and (2). The maximal components ofthe supp(I, µ) through x are described by the intersections

⋂D∈A D where A ∈ R(x). The maximal locus

of ρ determines at most o one maximal component of supp(I, µ) through each x.

After the blow-up at the maximal locus C = xi1 = . . . = xil = 0 of ρ, the ideal I = (xa1,...,ak) is equal toI ′ = (x′a1,...,aij−1,a,aij+1,...,ak) in the neighborhood corresponding to xij , where a = ai1 + . . .+ ail − µ < aij .In particular the invariant ordx(I) drops for all points of some maximal components of supp(I, µ). Thus themaximal value of ordx(I) on the maximal components of supp(I, µ) which were blown up is bigger than themaximal value of ordx(I) on the new maximal components of supp(I, µ). The algorithm terminates after afinite number of steps.


3.1. Summary of the resolution algorithm. The resolution algorithm can be represented by the follow-ing scheme.

Step 2. Resolve (I, µ).

Step 2a. Reduce (I, µ) to the monomial marked ideal I =M(I).

⇓

If I 6=M(I), decrease the maximal order of the nonmonomial part N (I) by resolving the companion

ideal O(I, µ).

Step 1. Resolve the companion ideal (J , µJ ) := O(I, µ) :

Replace J with J := C(H(J )) ≃ J . (*)

Step 1a. Move apart all strict transforms of E and supp(J , µ).

⇓

Move apart all intersections Hsα of s divisors in E

(where s is the maximal number of divisors in E through points in supp(I, µ)).

m

For any α, resolve J |(⋃

αHs

α).

Step 1b If the strict transforms of E do not intersect supp(J , µ), resolve (J , µ).

m

Simultaneously resolve all J |V (u) , where V (u) is a hypersurface of maximal contact.

(Use the property of homogenization ([51]), and Kollar’s trick ([37]).

Step 2b. Resolve the monomial marked ideal I =M(I).

Remarks. (1) (*) The ideal J is replaced with H(J ) to ensure that the algprithm constructed in Step1b is independent of the choice of the tangent direction u.

We replace H(J ) with C(H(J )) to ensure the equalities supp(J|S) = supp(J )∩S, where S = Hsα

in Step 1a and S = V (u) in Step 1b.(2) If µ = 1 the companion ideal is equal to O(I, 1) = (N (I), µN (I)) so the general strategy of the

resolution of (I, µ) is to decrease the order of the nonmonomial part and then to resolve the monomialpart.

(3) In particular if we desingularize Y we put µ = 1 and I = IY to be equal to the sheaf of the subvarietyY and we resolve the marked ideal (X, I, ∅, µ). The nonmonomial part N (Ii) is nothing but theweak transform (σi)w(I) of I.

In the next sections we provide a complexity bound for the algorithm.

4. Complexity bounds on a blow-up

Our purpose for the rest of the paper is to estimate the complexity of the desingularization algorithmdescribed in the previous sections.

4.1. Preliminary setup.

4.1.1. Affine marked ideals. An input of the algorithm is an affine marked ideal that is a collection of tuples

T := (Xα,β, Iα,β , Eα,β , Uα,β, (Cnα)α | α ∈ A, β ∈ Bα, µ),

where

(1) (Cnα)α ≃ Cnα

(2) Uα,β | β ∈ Bα is an open cover of (Cnα)α.(3) Uα,β ⊂ (Cnα)α is an open subset whose complement is given by fα,β = 0.


(4) Xα,β ⊂ (Cnα)α is a closed subset such that Xα,β ∩ Uα,β is a nonsingular m-dimensional variety(possibly reducible). Moreover there exists a set of parameters (coordinates) on Uα,β,

uα,β,1, . . . , uα,β,nα∈ C[xα,1, . . . , xα,nα

],

such that uα,β,i is a coordinate xα,j describing exceptional divisor or it is transversal to theexceptional divisors (over Uα,β) , and, moreover IXα,β

= (uα,β,i1 , . . . , uα,β,ik) ⊂ C[xα,1, ..., xα,nα],

for certain subset i1, . . . , ik ⊂ 1, . . . , nα.(5) Iα,β = 〈gα,β,1, . . . , gα,β,j〉 ⊂ C[xα,1, . . . , xα,nα

] is an ideal,

(6) Eα,β is a collection of smooth divisors in (Cnα)α described by some xα,j = 0 where j = s, s+1, . . . , nfor some 0 ≤ s ≤ m− nα.

(7) There exist birational maps iα1β1,α2,β2 : Xα1,β1 99K Xα2,β2 given by

Xα1,β1 ∋ x 7→ iα1β1,α2,β2(x) = (vα1β1,α2,β2,1

wα1β1,α2,β2,1, . . . ,

vα1β1,α2,β2,nα2

wα1β1,α2,β2,nα2

)(x) ∈ Xα2,β2

for regular functions vα1β1,α2,β2,1, . . . , vα1β1,α2,β2,nα2, wα1β1,α2,β2,1, . . . , wα1β1,α2,β2,nα2

∈ C[x1, . . . , xnα1]

(8) The birational maps iαβ,α′,β′ determine uniquely up to an isomorphism a variety XT in the followingsense: There exist open embeddings jα,β : Xα,β ∩ Uα,β → XT defining an open cover of XT , andsatisfying

j−1α2,β2

jα1,β1 = iα1β1,α2,β2

(9) µ ≥ 0 is an integer.(10) supp(Iα,β , µ) ∩ Uα,β ∩ Uα,β′ = supp(Iα,β′ , µ) ∩ Uα,β ∩ Uα,β′

Remarks. (1) The objects Xα,β, Eα,β , Iα,β , as well as corresponding functions gα,β,i, uα,β,j, xα,i arerelevant for the algorithm after they are restricted to Uα,β. Their behavior in the complement(Cnα)α \ Uα,β has no relevance.

(2) The operation of restricting to the maximal contact leads to considering open subsets Uα,β ⊂ (Cnα)α.(3) While studying the complexity of the algorithm we shall assume that the coefficients of the input

polynomials belong just to Z. All general considerations are given for coefficients from C, and remainvalid for any algebraically closed field of zero characteristic.

(4) The open embeddings jα,β can be constructed from iα1β1,α2,β2 after performing the algorithm butwe don’t dwell on it.

Definition 4.1.1. By the support of T we mean the collection of the sets

supp(T ) := supp(Xα,β ∩ Uα,β, Iα,β , Eα,β ∩ Uα,β ∩Xα,β, µ)α∈A,β∈B

Definition 4.1.2. Given an affine marked ideal T := (Xα,β , Iα,β, Eα,β , Uα,β, (Cnα)α | α ∈ A, β ∈ Bα, µ),

we say that an affine marked ideal T ′ := (Xα′,β′ , Iα,β , Eα,β , Uα,β, (Cnα)α | α ∈ A′, β ∈ B′

α, µ), is definedover T , provided

(1) There exist maps of set of indices p : A′ → A , and pα′ : B′α′ → Bp(α′)

(2) The canonical projection on the first α = p(α′) components

π′α : (Cnα′ )α′ → (Cnα)α

determine birational morphisms πα′,β′ := π′|Xα′,β′

: Xα′,β′ → Xα,β commuting with iα1β1,α2,β2 and

iα′1β

′1,α

′2,β

′2.

(3) There exist natural birational morphisms XT ′ → XT commuting with jα,β , and jα′,β′ .

We introduce the following functions to characterize the affine marked ideal

T := (Xα,β, Iα,β , Eα,β , Uα,βα,β, (Cnα)α, µ) :

(1) m(T ) = dim(XT ),(2) µ(T ) = µ,(3) d(T ) is the maximal degree of all polynomials in

Ψ(T ) := uα,β,i, gα,β,i, fα,β , vα,β,α2,β2,i, wα,β,α2,β2,j.

(4) n(T ) = maxnα,(5) l(T ) is the maximal number of all polynomials in Ψ(T ).


(6) q(T ) is the number of neighborhoods Uα,β in T ( i. e the number of the indices s.t α ∈ A, β ∈ B).(7) b(T ) the maximum of bit size of any (integer) coefficient of each of the polynomials in Ψ(T )

Remark. The function b(T ) is used only for the estimation of the total complexity of the algorithm. Inparticular it has no relevance for the estimates of the number of blow-ups, the maximal embedding dimensionn(T ), or the number of neighborhoods.

Algorithmically the input is represented by the coefficients of polynomials describing an affine markedideal T0. We assume

m(T0) = m, d(T0) ≤ d0, n(T0) ≤ n0, l(T0) ≤ l0, b(T0) ≤ b0.

Then in particular, the total bit-size of the input does not exceed b0 · l0 · dO(n0)0 , cf. [26].

4.1.2. Resolution of singularities. For simplicity consider an irreducible affine variety Y ⊂ Cn described bysome equations. The algorithm resolves Y by the following procedure:

Step A Find the generators of IY = 〈g1, . . . , gj〉 ⊂ C[x1, . . . , xn] and construct affine marked ideal

T := (X = Cn, IY , E = ∅, U = Cn,Cn, µ = 1)

Step B Start the resolving procedure for the affine marked ideal (X = Cn, IY , E = ∅, U = Cn,Cn, µ =1). (see below)

Step C Pick a nonsingular point p ∈ Y ⊂ Cn. Stop the resolution procedure when the constructedcenter of the following blow-up in the algorithm passes through the inverse image of p. As an output of theresolution algorithm we get an affine marked ideal

T ′ := (Xα′,β′ , Iα′,β′ , Eα′,β′ , Uα,β,Cnα′α′,β′ , 1)

over T . In particular we have a collection of projections

πα′ : (Cnα′ )α′ → Cn

(projection on the first n coordinates.). (Note that the restriction of πα defines a birational morphismπα′,β′ : Xα′,β′ → X which is an isomorphism in a neighborhood of p ∈ X .)

The center of the following blow-up is described on some open subcover Uα′,β′′α′∈A′,β′′∈B′′α′

of

Uα′,β′α′∈A′,β′∈B′α′

by Cα′,β′′ ∩ Uα′,β′′ for the closed subsets Cα′,β′′ ⊂ (Cnα′ )α′ . Consider the unique

irreducible component Cα′,β′′ of Cα′,β′′ containing the inverse image of the point p. Then

πα′,β′′ := π|Cα′,β′′∩Uα′,β′′: Yα′,β′′ := Cα′,β′′ ∩ Uα′,β′′ → Y

is a local resolution of Y . The resolution space Y is described by an open cover Yα′,β′′ , πα′,β′′α′,β′′ . The

sets Yα′,β′′ are represented as closed subsets of open subset Uα′,β′′ ⊂ Cnα′ .

4.1.3. Principalization. Given a smooth affine variety X ⊂ Cn, described by equations uX,i = 0, and anideal I = (g1, . . . , gk) on X and on Cn.

Step A First the algorithm finds affine neighbourhoods Uα,β in Cn each given by an inequality fα,β 6= 0in which X is represented by a family of local parameters

u1 = · · · = un−m = 0.

Moreover it finds the coordinates xi on Cn, such that (up to index permutation)

u1, · · · , un−m, xn−m+1, . . . , xn

is a complete set of parameters.

The local parameters ui are chosen among the input polynomials. To this end the algorithm can for eachchoice of i1, . . . , in−m ⊂ 1, . . . , i pick a non-vanishing identically minor fα,β of the Jacobian matrix ofuii.

We construct an affine marked ideal given by input tuple by

T := (Xβ := X, Iβ := I, Eβ = ∅, Uβ,Cnβ , µ = 1)

Step B The algorithm resolves T = (Xβ := X, Iβ := I, Eβ = ∅, Uβ,Cnβ, µ = 1). As an output we get

T ′ := (Xα′,β′ , Iα′,β′ , Eα′,β′ , Uα′,β′ , (Cnα′ )α′α′,β′,, 1),

over T .


Step C The variety X ′ := XT ′ is described by an open cover Xα′,β′ ∩ Uα′,β′α′,β′ for closed subsetsXα′,β′ ⊂ Cnα′ , and open subsets Uα′,β′ ⊂ Cnα′ (see (8) from 4.1.1). Moreover, we have a collection ofbirational morphisms πα′,β′ : Xα′,β′ ∩ Uα′,β′ → X ⊂ Cn. The principal ideal on Xα′,β′ is generated by

(g1 πα′,β′ . . . , gk πα′,β′) = (xa11 . . . x

anα′

nα′ )

4.2. Description of blow-up. Consider an affine marked ideal T := (Xα,β, Iα,β , Eα,β , Uα,β, (Cnα)α | α ∈

A, β ∈ Bα, µ) corresponding to a marked ideal (X, I, E, µ). Let C ⊂ X be a smooth center described asfollows:

We assume that there is an open subcover cover Uα,β′α∈A,β′∈B′α⊂ (Cnα)α ≃ Cn of Uα,β , together with

map of indices ρ : B′α → Bα, and the collection of the closed subvarieties Cα,β′ ⊂ (Cnα)α (of dimension

kαβ′ ≤ m), such that

(1)⋃

ρ(β′)=β Uα,β′ = Uα,β

(2) Cα,β′ ∩ Uα,β′ ⊂ supp(Iα,β′,µ) ∩ Uα,β′

(3) Cα,β′ is described on each Uα,β′ by the set of local parameters

uα,β′,1, . . . , uα,β′,nα−m, uαβ′,nα−m+1, . . . , uα,β′,nα−kαβ′ ∈ C[x1, . . . , xnα], i. e.

uα,β′,1 = . . . = uα,β′,nα−m = uαβ′,nα−m+1 = . . . = uα,β′,nα−kαβ′ = 0

where Xα,β is described on Uα,β ⊃ Uα,β′ by uα,β′,1 = . . . = uα,β′,nα−m = 0,(4) uα,β′,1, . . . , uα,β′,nα−kαβ′ are transversal to the exceptional divisors (over Uα,β′), or coincide with

coordinate functions describing the exceptional divisors.

Denote by

T ′ := (Xα′,β′ , Iα′,β′ , Eα′,β′ , Uα′,β′ , (Cnα′ )α′ | α′ ∈ A′, β′ ∈ B′α′, µ)

the resulting affine marked ideal obtained from T by the blow-up with the center C. Below we describemore precisely the ingredients of T ′.

The open cover after blow-up.

The blow-up creates a new collection of ambient affine spaces (Cnα′ )α′ . Namely, we can associate withfunctions uα,β′,i on (Cnα)α, where i = 1, . . . , nα − kαβ′ , the nα − kαβ′ affine charts

(Cnα′ )α′ , where α′ := (α, i), i = 1, . . . , nα − kαβ′ , nα′ := 2nα − kαβ′

We also create new collection of open subsets Uα′,β′ ⊂ (Cnα′ )α′ by taking the inverse images of Uα,β′ ⊂(Cnα)α under the morphisms (Cnα′ )α′ → (Cnα)α.

The birational maps

The natural projection πα : (Cnα′ )α′ → (Cnα)α on the first nα components defines the birational morphismπα′,β′ = πα|Xα′,β′ : Xα′,β′ → Xα,β′ , for any α, β′, such that Xα,β′ 6= ∅. This defines birational morphisms

iα′1β

′1,α

′2,β

′2

: Xα′1,β

′1

πα′1,β′

1→ Xα1,β′1

iα1β′1,α′

2,β2

99K Xα2,β′2

π−1

α′2,β′

2← Xα′2,β

′2.

Consider a blow-up XT ′ of C ⊂ XT . There exist open embeddings jα′,β′ : Xα′,β′ ∩Uα′,β′ → XT ′ inducedby jα,β : Xα,β′ ∩ Uα,β′ → XT , defining an open cover of XT ′ , and satisfying

j−1α′

2,β′2 jα′

1,β′1

= iα′1β

′1,α

′2,β

′2

Equations of blow-up.

Without loss of generality the blow-up in each of nα−kαβ′ affine charts (Cnα′ )α′ , where α′ := (α, i), i =1, . . . , nα − kαβ′ , can be described as follows: (For simplicity drop α, β indices below.)

Assume that the function ui0 , i0 ≤ n− k, defines the chart of the blow-up. The blow-up of Cn is a closedsubset bl(Cn) of C2n−k described by the following equations

uj − ui0xj+n = 0 for 0 < j ≤ n− k, j 6= i0

ui0 − xi0+n = 0

The exceptional divisors.


The exceptional divisor for this blow-up is given by ui0 = 0 on bl(Cn). Since ui0 = xi0+n we may representit by the coordinate xi0+n on C2n−k.

The previous exceptional divisors keep their form xj = 0 if they do not describe C, or they convert toxj+n(= uj/ui0) if they were described by the function uj ≡ xj .

The strict transform of X.

Recall that X is described by u1 = . . . = un−m = 0 on U ⊂ Cn. The blow-up of X = Xα,β is a closedsubset X ′ ⊂ C2n−k which is described by a new set of equations:

(1) uj − ui0xj+n = 0 for 0 < j ≤ n− k, j 6= i0(2) ui0 − xi0+n = 0(3) xj+n = 0 for 0 < j ≤ n−m, j 6= i0.

In some situations we consider additionally the induced equation(4) 1 = 0 if 0 < i0 ≤ n−m

(Note that the equations of the first two types describe the blow-up bl(Cn) of Cn. The third and the fourthtypes of the equations xj+n = uj/ui0 = 0, j 6= i0 (or 1 = ui0/ui0 = 0 ) describe the strict transform ofX inside bl(Cn). In the latter case if 0 < j = i0 ≤ n − m the strict transform is an empty set in therelevant chart. Still we shall keep the uniform description of the objects and their transformations, and donot eliminate any equations in the description of the empty set.)

4.2.1. The generators of Iα,β after blow-up. We will not compute the controlled transforms of the generatorsof I (= Iα,β) (over U) directly. Instead we modify them first. The generators gi of I satisfy, by Lemma2.2.1, the condition

gi · fri ∈ IµC + IX ,

where V (f) = Cn \ U ⊂ Cn (we dropped indices α, β here).

For any generator gi write

gi · fri =

∑

an−m+1+...+an−k=µ

h(an−m+1,...,an−k),iuan−m+1

n−m+1 · . . . · uan−k

n−k +∑

j:=1...,m−n

hijuj

Set a := (an−m+1, . . . , an−k), ua := uan−m+1

n−m+1 · . . . · uan−k

n−k . Then we can rewrite above as

gi · fri =

∑

|a|=µ

haua +

∑

j:=1...,m−n

hijuj

To bound ri and deg(ha,i), deg(hij) we first consider a similar equality

gi · fRi =

∑

|a|=µ

Haua +

∑

j:=1...,m−n

Hijuj

for certain Ri, Ha,i, Hij . We introduce a new variable z and we get an equality

gi = zRi · (∑

|a|=µ

Haua +

∑

j:=1...,m−n

Hijuj) + gi · (∑

0≤j≤r−1

(z · f)j) · (1− z · f),

in other words gi belongs to the ideal generated by ua, uj, 1− z · f . Therefore one can represent

gi =∑

|a|=µ

Ha,i · ua +

∑

j:=1...,m−n

Hij · uj + H · (1 − z · f)

for suitable polynomials Ha,i, Hij , H with degrees less than (d · µ)2O(n)

due to [43], [24], [40]. Hence sub-

stituting in the latter equality z = 1/f and cleaning the denominator we obtain the bound (d · µ)2O(n)

onri, deg(ha,i), deg(hij).

The generators after blow-up and their degree.

Using the above we can describe the controlled transform of I in terms of controlled transforms of itsmodified generators. Define the modified generators of I to be

gi =∑

|a|=µ

haua

Then their controlled transforms are given by

σc(gi) = u−µn−kσ

∗(ha,iua)


Denote by G(d, n, µ) the bound on the degree of the resulting marked ideal T ′ after a blow up applied toa marked ideal T , provided that d(T ) ≤ d, n(T ) ≤ n. Thus, by the above:

Lemma 4.2.1. G(n, d, µ) < (d · µ)2O(n)

.

4.3. Elementary operations and elementary auxillary functions. To estimate the complexity of thedesingularization algorithm we introduce few auxiliary functions related to the ingredients of T . It isconvenient to associate with T with data (m, d, n, l, q, µ) the vector

γ := (r,m, d, n, l, q, µ) ∈ Z7≥0,

where r is the subscript of the element of the resolution (Tr)r=0,1,...,

4.3.1. The effect of a single blow-up. Summarizing the above we get the following

Lemma 4.3.1. Consider the object T := (Xα,β, Iα,β , Eα,β , Uα,βα,β, n, µ) with data (m,µ, d, n, l, q). LetT ′ := (X ′

α′,β′ , I ′α′,β′ , E′α′,β′ , U ′

α′,β′α′,β′ , nα′ , µ) denote the object with data (m,µ, d′, n′, l′, q′) obtained from

T by a single blow-up at the center C represented by the collection of closed sets Cαβ′′ ⊂ (Cnα)α describingthe center in open subsets Uαβ′′ ⊂ Uαβ. Assume that the maximal degree of of the polynomials describingcenter is less than d. Assume that q gives also a bound for the number of open neighborhoods Uαβ′′ . Then

(1) d′ ≤ G(n, d, µ) < (d · µ)2O(n)

.(2) n′ ≤ 2n(3) l′ ≤ l + n(4) q′ ≤ n · q

The effect of the single blow-up can be measured by the function.

Bl(r,m, d, n, l, q, µ) := (r + 1,m,G(n, d, µ), 2n, l + n, n · q, µ)

The multiple effect of the t blow-ups can be measured by the recursive function.

By

Bl(r,m, d, n, l, q, µ, t) = Bl Bl(r,m, d, n, l, q, µ, t− 1)

or shortly

Bl(γ, t) = Bl(Bl(γ, t− 1))

Note that

Bl(r,m, d, n, l, q, µ, t) = (r + t,m,G(n, d, µ, t), 2tn, l + 2t−1n, (2t+1 − 1) · ntq, µ)

for the relevant function G(n, d, µ, t).

5. Bounds on multiplicities and degrees of coefficient ideals

5.0.2. The maximal multiplicity of I on the subvariety X ⊂ Cn. Let d be the maximal degree of (X, I) onCn. Denote by M(d, n) a bound on the multiplicities of ideals I on X . To verify a bound on M(d, n) we mayassume (after a linear transformation of the coordinates) that the order of the polynomial uij−xj is at least 2for all 1 ≤ j ≤ n−m. For any polynomial g ∈ Iα,β one can find polynomials hj ∈ C[x1, . . . , xn], 0 ≤ j ≤ n−mand h ∈ C[xn−m+1, . . . , xn] such that

h0 · g +∑

1≤j≤n−m

hj · uij = h(xn−m+1, . . . , xn).(5)

Moreover, rewriting the latter equality over the field C(xn−m+1, . . . , xn) in the form

h0 · g +∑

1≤j≤n−m

hj · uij = 1,(6)

where hj = hi

h ∈ C(xn−m+1, . . . , xn)[x1, . . . , xn−m], with the common denominator in C[xn−m+1, . . . , xn]for 0 ≤ j ≤ n−m.


We apply to (6) the Effective Nullstellensatz (see e.g. [15], [23], [36], [34]). This gives us the bound dO(n)

on the degrees of hi = hi/h with respect to variables x1, . . . , xn−m (for certain solutions) . To find hi/h onecan solve the latter equality considering it as a linear system over the field C(xn−m+1, . . . , xn).

The algorithm can find

hj =∑

aI,jxI

with indeterminates aI,j ∈ C(xn−m+1, . . . , xn), and monomials xI = xi11 · . . . · x

in−m

n−m with degrees i1 + . . . +

in−m ≤ dO(n) substituting hj in (6) and solving linear system over the field C(xn−m+1, . . . , xn). Clearingthe common denominator in (6) gives (5) with

deg(h), deg(hj) ≤ dO(n2), 1 ≤ j ≤ n−m

Now we have to estimate the maximal multiplicity ordx(g|X) for g ∈ C[x1, . . . , xn], such that deg(g) ≤ dand x ∈ X . We use (5).

We get immediately by above

Lemma 5.0.2. The maximal multiplicity ordx(g|X) on X for any function g ∈ C[x1, . . . , xn], such that

deg(g) ≤ d and x ∈ X is bounded by the function M(d, n) = dO(n2) constructed as above.

ordx(g|X) ≤M(d, n)

Proof.

ordx(g|X) ≤ ordx(h0·g)|X = ordxh(xn−m+1, . . . , xn)|X = ordxh(xn−m+1, . . . , xn) ≤ deg(h) ≤M(d, n) = dO(n2).

5.0.3. Derivations on the subvariety X ⊂ Cn. In order to follow the construction of the algorithm weuse the language of derivations DerX on X . Since our X is embedded into Cn it is natural to representall objects on X as the restriction of the relevant objects on Cn to X ⊂ Cn. Unfortunately the sheaf ofderivations on Cn does not restrict well to X :

DerCn|X 6= DerX .

Instead we consider

DerCn,X := D ∈ DerCn | D(IX) ⊂ (IX)

Lemma 5.0.3. Let u1 = 0 . . . , un−m = 0 be describe X ⊂ Cn. Then the ring DerCn,X is generated by

ui · duj1≤i≤n−m, 1≤j≤n−m ∪ duj

n−m<j≤n.

In particular the restriction DerCn,X|X = DerX is generated by

dujn−m<j≤n.

Proof. Follows form the definition.

Note that since we have

(duj)j = ((∂uj/∂xi)i,j)

−1 · (dxi)i =

1

det((∂uj/∂xi)i,j)adj((∂uj/∂xi)i,j) · (dxi

)i

we get immediately

Lemma 5.0.4. Let U := x ∈ Cn | det((∂uj/∂xi)i,j) 6= 0.

The sheaf DerCn,X is generated over U ⊂ Cn by

ui · d′uj1≤i≤n−m, 1≤j≤n−m ∪ d

′ujn−m<j≤n,(7)

where

(d′uj)j = adj((∂uj/∂xi)i,j) · (dxi

)i

The derivations (7) generate a subsheaf DerCn,X of DerCn,X over Cn. Both sheaves coincide over U . Thus

we shall replace DerCn,X with DerCn,X for our computations over U .


Lemma 5.0.5. Let I be any ideal on Cn. Assume the maximal degree of some generating set of I is ≤ d1,and the maximal degree of ui is less than d2 then the maximal degree of generators of DerCn,X(I) is boundedby d1 + nd2.

5.0.4. Construction of the coefficient homogenized companion ideal. Recall then in the step 2 for the markedideal (I, µ), we find the maximal multiplicity µ ≤ M(d, n) and construct companion ideal O(I), for whichimmediately we take homogenized coefficient ideal J := C(H(O(I)))). In our situation of the set X ⊂ Cn

defined by set of parameters ui1≤i≤n−m on the open set U ⊂ Cn we shall use DerCn,X instead of DerX forthe above devinition. Denote the relevant operations of homogenization and coefficient ideal by H(), andC(). Immediately from the definition we get the formula for a bound A(d, n, µ) on the degrees of generatorsof the marked ideal C(H(Iq)). Note, first,that we have the the following bounds on the multiplicities:

Lemma 5.0.6. µ(N (I)) = µ, µ(O(I)) ≤ µ · µ and µ(J ) ≤ (µ · µ)! ≤ (µ ·M(d, n))!

As a Corollary we obtain

Lemma 5.0.7. The maximal degree of generators of

J := C(H(O(I)))) is bounded by A(d, n, µ) := (µ · µ)!nd ≤ (µ ·M(d, n))!nd ≤ (dO(n2))!.

Proof. Follows from Lemma 5.0.5

5.0.5. Restriction to hypersurface of maximal contacts, and to exceptional divisors. In the step1 we restrict I to intersections of the exceptional divisors and maximal contact.

We need to estimate a bound B(d, n, µ) for the degree of the maximal contact u ∈ DerCn,Xµ−1

(N (I)). Itfollows immediately form Lemma 5.0.5 that

Lemma 5.0.8. The maximal degree of any maximal contact u ∈ DerCn,Xµ−1

(N (I)) is bounded by

B(d, n, µ) := µnd ≤M(d, n) · nd ≤ dO(n2).

5.0.6. The bound for the number of generators of J . First, state the basic properties for the number ofgenerators of the ideal in the lemma:

Lemma 5.0.9. (1) The number of generators of DerCn,X(I) is given by (n + 1)l(I).

(2) The number of generators of DerCn,Xi(I) is given (n + 1)il(I)

(3) The number of generators of Ii is bounded by l(I)i

(4) The number of generators of I ′ = O(I) can be bounded by LO(l(I), µ) := l(I)µ + 1.

(5) The number of generators of I ′ = H(I, µ) can be bounded by LH(l(I), µ) := µ(n + 1)µ2

l(I)µ.(6) The number of generators of I ′ = C(I, µ) can be bounded by LC(l(I), µ)) := µ(n + 1)µ!l(I)µ!.

Proof. Immediately from the definition.

By the Lemma we getl(H(O(I))) ≤ LH(LO(l(I), µ), µ ·M(d, n))

l(J ) = l(C(H(O(I)))) ≤ LC(LH(LO(l(I), µ), µ ·M(d, n))), µ ·M(d, n))

Thus we get

Corollary 5.0.10. l(J ) ≤ F (d, n, µ), where

F (d, n, µ) := LC(LH(LO(l(I), µ), µ ·M(d, n))), µ ·M(d, n))

Remark. The algorithm of [12] does not involve the homogenization step and therefore gives better estimatesfor the introduced elementary functions. In particular,

-The degree of generators of J is bounded by B(d, n, µ) (which improves the bound A(d, n, µ), cf. Lemma5.0.7),

- The number of generators l(J ) can be bounded by LC(LO(l(I), µ), µ ·M(d, n)) (which improves thebound F (d, n, µ), cf. Corollary 5.0.10),

However, the above improvements do not affect overall Grzegorczyk complexity class Em+3. (See Theorem6.4.2.)

Summarizing,


Lemma 5.0.11. The effect of passing from I to J = (C(H(O(I)))) as in the Step 2a/Step1 can be describedby the function

∆2a(r,m, d, n, l, q, µ) := (r,m,A(d, n, µ), n, F (d, n, µ, l), q, (µ ·M(d, n))!)

5.0.7. The bound for the number of maximal contacts and the relevant neighborhoods. We shall constructmaximal contacts along with the open neighborhood for which it is defined. Each maximal contact we

consider u ∈ DerCn,Xµ−1

(N (I)) is of the form u = Dµ−1(gi), where Dµ−1 = Da11 . . . Dan

n is a certaincomposition of µ− 1 differential operators (7). (i.e. Di are of the from as in (7), and a1 + . . .+ an = µ− 1 .

Consider all differential operators Dµr r∈R which are certain compositions of µ differential operators (7)

and take all the corresponding functions fr,i := Dµr (gi). On the open set Ur,i = U \ V (fr,i) consider the

maximal contact ur,i = Dµ−1(gi), where Dµ−1 = Da11 . . . Dan

n is a certain composition of µ− 1 differential

operators (7) obtained from Dµr = Db1

1 . . . Dbnn by replacing one of the positive bi with bi − 1 (i.e ai := bi− 1

for some bi > 0 and aj = bj for j 6= i.)

Lemma 5.0.12. The number of the maximal contacts ur,i ∈ DerCn,Xµ−1

(N (I)) and at the same time thenumber of neighborhoods Ui,r ⊂ U can be bounded by

C(d, n, µ) · l(I),

where l(I) is the number of generators of I, and

C(d, n, µ) :=

(M(d, n) + n

n

)

Proof. The number of the maximal contacts is bounded by(µ+nn

)· l(I) ≤ C(d, n, µ) :=

(M(d,n)+n

n

)

Summarizing,

Lemma 5.0.13. The effect of passing from I to J = (C(H(O(I)))) and then to J|Hsin Step 1a or to J|V (u)

as in the Step 1b or can be described by the function

∆1(r,m, d, n, l, q, µ) := (r,m− 1, A(d, n, µ), n, F (d, n, µ, l), q · C(d, n, µ), (µ ·M(d, n))!)

Note that the restriction to the maximal contact does not affect the degree since the function B(d, n, µ)measuring the degree of maximal contact is smaller than A(d, n, µ)

Remark. A particular form of the obtained bounds does not influence much on the Theorem 6.4.2: we needonly that the functions belong to class E3. (See the beginning of the next section.)

6. Complexity bound of the resolution algorithm in terms of Grzegorczyk’s classes

6.1. Language of Grzegorczyk’s classes. The complexity estimate of the desingularization algorithmwhich we provide in this Section is given in terms of the Grzegorczyk’s classes E l, l ≥ 0 [27], [50] of primitive-recursive functions. For the sake of self-containdness we provide the definition of E l by induction on l(informally speaking, E l consists of integer functions Zs → Zt whose construction requires l nested primitiverecursions).

For the base of definition E0 contains constant functions xk 7→ c, functions xk 7→ xk + c and projections(x1, . . . , xn) 7→ xk for any variables x1, . . . , xn.

Class E1 contains linear functions xk 7→ c · xk and (xk1 , xk2) 7→ xk1 + xk2 .

For the inductive step of the definition assume that functions G(x1, . . . , xn), H(x1, . . . , xn, y, z) ∈ E l.Then function F (x1, . . . , xn, y) defined by the following primitive recursion

F (x1, . . . , xn, 0) = G(x1, . . . , xn)(8)

F (x1, . . . , xn, y + 1) = H(x1, . . . , xn, y, F (x1, . . . , xn, y))(9)

belongs to E l+1.


To complete the definition of E l, l ≥ 0 take the closure with respect to the composition and the followinglimited primitive recursion:

let G(x1, . . . , xn), H(x1, . . . , xn, y, z), Q(x1, . . . , xn, y) ∈ E l, then function F (x1, . . . , xn, y) defined by(8),(9) also belongs to E l, provided that F (x1, . . . , xn, y) ≤ Q(x1, . . . , xn, y).

Clearly, E l+1 ⊃ E l.

Observe that E2 contains all the polynomials with integer coefficients.

E3 contains all the towers of exponential functions.

E4 contains all tetration functions [27], [50].

The union ∪l<∞E l coincides with the set of all primitive-recursive functions.

6.2. Resolution algorithm as a graph. It is instructive to represent the resolution algorithm in a formof a tree T as in the following Figure.

2

1

2a

2a

1

2b

1

2

1a2

1a

1b

1b

2

2

1a

1a 2

2a1

2a

2b

M

2a

2bM

1

1

2

2

Figure 1.

Each node a of T corresponds to an intermediate object Ta = (Xα,β, Iα,β , Eα,β , Uα,βα,β, n, µ). Eachnode a is labeled either by 1 or 2 depending on whether it corresponds to step 1 or 2 in the description ofthe algorithm (see the previous sections). An edge from a node labeled by 1 leads to its child node labeledby 2 and the edge is labeled in its turn either by 2a or by 2b depending on to which step it corresponds.Similarly, an edge from a node labeled by 2 leads to its child node labeled by 1 and is labeled in its turneither by 1a or by 1b. In the Figure a child node is always located to the right from a node.

The algorithm yields T by recursion starting with its root. Assume that a and Ta is already yielded. Thenext task of the algorithm is to resolve object Ta. To this end the algorithm first constructs the child nodesa1, . . . , at of a according to the algorithm. The order of yielding a1, . . . , at goes from up to down in theFigure. The algorithm resolves objects Ta1 , . . . , Tat−1 by recursion on 0 ≤ t ≤ t and in the process modifiesTa := Ta(0) obtaining current object Ta(t− 1). Then the algorithm yields at, Tat

, resolves Tatand collects

all the blow ups produced while resolving Tatand applies them (with the same centers) to the current object

Ta(t− 1), the resulting object we denote by Ta(t). This allows the algorithm to yield a child node at+1 andTat+1 following the description from the previous sections.


For the leaves of T there are two possibilities: either a leaf is labeled by 2 or a certain node a labeled by2 could have a single edge (the lowest in the Figure among the edges originating at a) labeled by 2b whichleads to a child node of a being a leaf corresponding to the monomial case (labeled by M). Note also thatif a is labeled by 1 then few top edges originating at a are labeled by 1a and the remaining bottom ones arelabeled by 1b (in the order from up to down in the Figure).

Observe that the dimension of varieties Xα,β corresponding to a drops while passing to any of its childnodes when a is labeled by 1, and the dimension does not increase when a is labeled by 2. Therefore, thedepth of T does not exceed 2m.

6.3. Main recursive functions. Now we proceed to the bounds of some recursive functions related to theingredients of T . Set

γ := (r,m, d, n, l, q, µ) ∈ Z7≥0

Let T∗ be the canonical resolution of T . For simpilicity of notation introduce following function definedon Z7

≥0:

Let

Γ(m)(γ) := (r + R(m)(γ),m,D(m)(γ), N (m)(γ), L(m)(γ), Q(m)(γ), µ).

(Here the subscript r can be interpreted as the subscript in the resolution of T ),

where

(1) R(m)(γ) the number of blow-ups needed to resolve the initial marked ideal with data bounded by(m, d, n, l, q, µ).

(2) D(m)(γ) be a function bounding the maximum of the degrees of all the polynomials which representT∗ and all objects constructed on the way, in particular the centers.

(3) N (m)(γ) as the bound for the dimensions of the ambient affine spaces constructed on the way, .(4) L(m)(γ) the bound for number of polynomials appearing in description of a single neighborhood Uα,β

upon resolving T .(5) Q(m)(γ) the number of neighborhoods in all the auxiliary objects, in particular the centers. appearing

upon resolving T .

Remark. The functions R(m)(γ), D(m)(γ), N (m)(γ) do not depend upon l, q.

6.3.1. Algorithm revisited. Let (I, µ) be a marked ideal on m-dimensional smooth variety X . Consider thecorresponding object

T (m) = (Xα,β, Iα,β , Eα,β , Uα,β,Cnα | α ∈ A, β ∈ B, µ)

with the initial data

γ := (0,m, d, n, l, q, µ).

Our next goal is two-fold. We will give recursive formula for Γ(m), R(m), D(m), N (m), L(m), Q(m) andprove by induction on m that functions Γ(m), and others belong to Grzegorczyk’s class Em+3. In the baseof induction for m = 0 functions

R(0) = 1, D(0) = O(dn), N (0) ≤ 2n, L(0) ≤ l · (dn)O(n), Q(0) ≤ nq

belong to class E3 as well as Γ(0).

Now we proceed to the inductive step. Assume that Γ(m−1), R(m−1), D(m−1), N (m−1), L(m−1), Q(m−1)

belong to Grzegorczyk’s class Em+2, and m ≥ 1.

If I = 0 then the resolution is done by the single blow-up at the center C = X and the object T (m) is

transformed into an object T (m)1 with X1 = ∅ and data bounded by Bl(γ) ∈ E3 (see Lemma 4.3.1).

If I 6= 0 the resolution algorithm can be represented by the following scheme.

Step 2. Resolve (I, µ) on m-dimensional smooth variety X . Consider the corresponding object T (m) withinitial data γ := (0,m, d, n, l, q, µ). Let µ denote the maximal order of N (I) on X . We have the followingestimate for µ:

µ ≤M(d, n) ∈ E3

(cf Lemma 5.0.2).


Step 2a. In this Step we are going to decrease the maximal order of the nonmonomial part N (I) byresolving the companion ideal O(I, µ). In fact we perform additional modification of O(I) and construct

the ideal J := C(H(O(I))). This corresponds to the new object T (m)1 with the initial data

γ(2a) := ∆2a(γ) ∈ E3,

(see Lemma 5.0.11.)

The object T(m)

1 will be then resolved and its resolution will cause the maximal order to decrease. Reso-

lution T (m)1 is done by performing Step 1.

Step 1. In this step we resolve J , i.e T(m)

1 . The Step splits in two Steps (1a) and (1b).

Step 1a. Move apart all unions of the intersections Hsα ⊂ Cn

α of s divisors in E, where s is the maximalnumber of divisors in E through points in supp(I, µ)). For any α, resolve all J |Hs

α. We construct new object

T(s)

2 := (Hsα,β, Iα,β , Eα,β , Uα,β,C

nα | α ∈ A, β ∈ B, µ),

with the initial data bounded by

γ(1a) := ∆1(γ) ∈ E3

with s ≤ m− 1. (See Lemma 5.0.13.)

By the inductive assumption, the resolution of T (s)2 , i.e the sequence T (s)

2∗ of the induced intermediate

objects determined by the blow-ups, requires at most R(m−1)(γ(1)) blow ups. The maximal degree of the

polynomials of the centers and the objects T (s)2∗ describing the resolution does not exceed D(m−1)(γ(1)). The

dimension n of the objects does not exceed N (m−1)(γ(1)).

Note that the resolution of T(s)

2∗ determines a multiple blow-up T(m)

1∗ of T(m)

1 consisting of R(m−1)(γ(1))

blow-ups. We have a direct correspondence between objects T (m)1∗ , and T (s)

2∗ . The bound

Γ(m−1)(∆1(γ)) ∈ Em+2,

for the data for the resolution T (s)2∗ , given by the induction, remains valid for data for T (m)

1∗ as we use the samecenters, the same ambient affine spaces and others for these multiple blow-ups. Only the strict transformsof the current X are different which does not affect the bounds for data. We have additional equations for

describing current X in T (s)2∗ comparing to those in T (m)

1∗ .

Step 1a is performed at most s ≤ m times. Introduce the auxillary unknown t = 0, 1, . . . ,m, and the

function Γ(m)1a (γ, t) which measures the possible effect after performing Step 1a t times.

Γ(m)1a (γ, 0) := ∆1(γ) ∈ E3

Γ(m)1a (γ, t + 1) := Γ(m−1)(Γ

(m)1a (γ, t))

Since the Step 1a is performed at most m times its final effect after completing Step 1a and passing toStep 1b is the measured by the function

Γ(m)1b (γ) := Γ

(m)1a (γ,m)

Note that for any fixed value of t = t0 (in particular, for t = m) functions Γ(m)1a (γ, t0), belongs to class Em+2

due to the inductive hypothesis and by virtue of Lemma 4.3.1, Corollary 5.0.10. Therefore Γ(m)1b belongs

to the class Em+2. (We use here the property that Grzegorczyk classes are closed under the composition.)After performing Step 1a we moved apart all strict transforms of E and supp(J , µ).

Step 1b If the strict transforms of E do not intersect supp(J , µ), we resolve (J , µ) i.e. the object T (m)1 .

This is achieved by resolving J |V (u) (by induction), where V (u) is a hypersurface of maximal contact. After

completing Step 1a the bound γ(1a) is transformed to

γ(1b) := Γ(m)1b (γ) = (r(1b),m, d(1b), n(1b), l(1b), q(1b), (µ · (M(d, n)))!)

(cf. Lemma 5.0.6.)

Passing (J , µ) to J |V (u) we adjoin the equations of maximal contact as well as create new neighborhoods.

This operation has been reflected in the construction of ∆1(γ). By the construction of Γ(m)1b (γ) and ∆1(γ)

the degree of the maximal contact does not exceed d(1b), while the number of neighborhoods does not exceed

q(1b). In other words Γ(m)1b (γ) bounds the initial data for J |V (u).


The resolution process J |V (u) leads eventually to the resolution of the object T (m)1 corresponding to J

with the data bounded by

Γ(m)1 (γ) := Γ(m−1)(Γ

(m)1b (γ)) = Γ

(m)1a (γ,m + 1) = (r(1),m, d(1), n(1), l(1), q(1), (µ · (M(d, n)))!),

for the relevant r(1), d(1), n(1), l(1), q(1).

Hence the function Γ(m)1 belongs to class Em+2 by inductive hypothesis and by virtue of Lemma 4.3.1 and

Corollary 5.0.10. This completes Step 1.

The object T (m) corresponding to I with initial data γ is transformed to the new object with the databounded by

Γ(m)2a (γ) := (r(1),m, d(1), n(1), l(1), q(1), µ)

with smaller µ – the maximal order of N (I). (Note that µ < M(d, n)).

This completes Step 2a. The Step 2a is then repeated at most M(d, n) times until the maximal orderdrops to zero as we arrived at the monomial case. The final effect of Step 2a is measured then by therecursive function

Γ(m)2a (γ, 0) = γ

Γ(m)2a (γ, t + 1) = Γ

(m)2a (Γ

(m)2a (γ, t)).

Therefore the function Γ(m)2a belongs to class Em+3 by definition of the Grzegorczyk’s classes (see (8),(9)),

and by virtue of Lemmas 4.3.1, 5.0.2.

Putting t = M(d, n) gives the final effect after completing all necessary Steps 2a and subsequent passingto the Step 2b

Γ(m)2b (γ) := Γ

(m)2a (γ,M(d, n)),

and thereby function Γ(m)2b belongs to class Em+3 as well.

The procedure eventually reduces (I, µ) to the monomial marked ideal I =M(I).

Step 2b. Resolve the monomial marked ideal I = M(I). The marked ideal corresponds to the objectT (m) with data

(r(2b),m, d(2b), n(2b), l(2b), q(2b), µ) := Γ(m)2b (γ).

The resolving of I = (xα) consists of blow-ups each of which decreases the multiplicity |xα| ≤ d(2b). Theresolution of I requires at most d(2b) blow-ups. Thus the final data solution can be bounded by the function

Γ(m)(γ) := Bl(Γ(m)2b (γ), d(2b)) ∈ Em+3

We summarize the achieved bounds in the following Corollary (recall that the notations one can find insubsection 4.1.1).

Corollary 6.3.1. While resolving a marked ideal (X, I, E, µ) on X ⊂ Cn by Hironaka algorithm the degreed, and the number l of the occurring polynomials, the embedding dimension n, the number r of the blow upsand the number q of the affine neighborhoods satisfy , for a fixed m = dim(X), the recursive equalities aboveand are majotated by a function

(r,m, d, n, l, q, µ) := Γ(m)(0,m, d0, n0, l0, q0, µ) ∈ Em+3,

for the initial values d = d0, n = n0, l = l0, q = q0,

6.4. Complexity of the algorithm. The principal complexity result of the paper states that

Theorem 6.4.1. While resolving a marked ideal (X, I, E, µ) on X ⊂ Cn by Hironaka algorithm its com-plexity can be bounded , for a fixed m = dim(X), by

bO(1) · F (m)(d0, n0, l0, q0, µ)

for a certain function F (m)(d0, n0, l0, q0, µ) ∈ Em+3.


Proof. Indeed, each step of the algorithm consists of solving a certain subroutine (basically, solving alinear system) over the coefficients of current polynomials. Therefore, Corollary 6.3.1 provides a bound onthe number of arithmetic operations with the coefficients of current polynomials (being a function from classEm+3). On the other hand, all the coefficients of the polynomials for the next step are obtained as results

of these arithmetic operations, so bit sizes of the coefficients grow at most as b · F(m)1 for a suitable function

F (m)1 ∈ Em+3. A cost of a single arithmetic operation is obviously polynomial.

As a corollary we obtain the following theorem.

Theorem 6.4.2. While

(1) resolving singularities of X ⊂ Cn0 , or(2) principalizing an ideal sheaf I on a nonsingular X ⊂ Cn0

by Hironaka algorithm the degree d, and the number l of the occurring polynomials, the embedding dimensionn, the number r of the blow ups and the number q of the affine neighborhoods satisfy , for a fixed m = dim(X),the recursive equalities above and are majotated by a function

(r,m, d, n, l, q, 1) := Γ(m)(0,m, d0, n0, l0, q0, 1) ∈ Em+3,

for the initial values d = d0, n = n0, l = l0, q = q0,

The complexity of the algorithm is bounded by

bO(1) · F (m)(d0, n0, l0, q0, µ),

for a certain function F (m)(d0, n0, l0, q0, µ) ∈ Em+3.

Remark. Above in the proof we gave a more explicit form of F (m) providing an additional information onits dependance on r, d, n, l, q, µ. But the main consequence of the Theorem is that m = dimX brings themost significant contribution into the complexity bound.

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E. Bierstone, Fields Institute, 222 College Street, Toronto, Ontario, Canada M5T 3J1 and Department ofMathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4

E-mail address: [email protected]

D. Grigoriev, Laboratoire des Mathematiques , Universite de Lille I, 59655, Villeneuve d’Ascq, Francehttp://en.wikipedia.org/wiki/Dima Grigoriev


P.D. Milman, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4


J. Wlodarczyk, Department of Mathematics, Purdue University, West Lafayette, IN-47907, USA


effective hironaka resolution and its complexity...

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